Directory UMM :Data Elmu:jurnal:J-a:Journal of Computational And Applied Mathematics:Vol102.Issue2.1999:

Journal of Computational and Applied Mathematics 102 (1999) 221–234

Estimations on numerically stable step-size for neutral delay
dierential systems with multiple delays
Guang-Da Hu a , Baruch Cahlonb; ∗
a

Department of Control Engineering, Harbin Institute of Technology, 150001, People’s Republic of China
b
Department of Mathematics and Statistics, Oakland University, Rochester, MI 48309-4485, USA
Received 3 May 1998; received in revised form 11 September 1998

Abstract
We derive two estimations of numerically stable step-size for systems of neutral delay dierential equations with multiple
delays. The stable step-size for numerical integration of NDDEs with multiple delays can be easily selected by means
of the logarithmic norm and the spectral radius of certain matrices. Both explicit linear multistep methods and explicit
c 1999 Elsevier Science B.V. All rights reserved.
Runge–Kutta methods are considered.
Keywords: Numerical stability region; Neutral delay dierential systems with multiple delays; Underlying numerical
methods for ODEs; Stable step-size

1. Introduction
Consider the system of neutral delay dierential equations (NDDEs) with multiple delays described
by
u(t)
˙ = f(t; u(t); u(t−1 ); : : : ; u(t−m ); u(t−
˙
˙
1 ); : : : ; u(t−
m ));
u(t) = g(t);

t¿0;

−m 6t60;

(1)

where f and g are given vector-valued functions, j is a given positive constant for j = 1; : : : ; m,
m ¿m−1 ¿ · · · ¿1 ¿0, and u(t) is the unknown vector-valued function.
We assume the existence of a unique solution of system (1). As in the case of ordinary dierential equations (ODEs), the stability of numerical solution of NDDEs is crucial in obtaining good

∗

Corresponding author.

c 1999 Elsevier Science B.V. All rights reserved.
0377-0427/99/$ – see front matter
PII: S 0 3 7 7 - 0 4 2 7 ( 9 8 ) 0 0 2 1 5 - 5

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G.-D. Hu, B. Cahlon / Journal of Computational and Applied Mathematics 102 (1999) 221–234

numerical approximations. As in the ODEs, the stability analysis is carried out through the linear
system of NDDEs with multiple delays, i.e.,
u(t)
˙ = Lu(t)+

m
X

[Mj u(t−j )+Nj u(t−
˙
j )];

t¿0;
(2)

j=1

u(t) = g(t);

−m 6t60:

Here L, Mj and Nj ∈ Cd×d are constant complex-valued matrices for j = 1; : : : ; m and m ¿m−1 ¿ · · ·
¿1 ¿0. Stability analyses of linear multistep methods and Runge–Kutta methods for system (2)
in the case j = j have been given in [5, 7]. For earlier results of numerical solutions of Neutral
equations and delay dierential equations with many delays, see [1, 10, 11]; for recent results on
numerical stability of Neutral and delay dierential equations, see [4, 6].
The goal of the present paper is to extend the study of [5, 8, 9] and to give two practical ways to
estimate the stable step-size for explicit linear multistep methods and explicit Runge–Kutta methods

applied to system (2).
2. Numerical stability of (2)
In this section, we will review the results of [6]. We denote by j (A) the jth eigenvalue of
A ∈ Cn×n (j = 1; 2; : : : ; n). Consider the matrix


Q(v1 ; : : : ; vm ) =I −

m
X
j=1


−1 

Nj vj  L+

m
X
j=1



Mj vj  ;

where mj=1 kNj k¡1; vj ∈ C and |vj |61. The following lemma states a sucient condition for delayindependent stability of system (2).
P

Lemma 2.1 (Hu and Hu [6]). System (2) is asymptotically stable if
m
X

kNj k¡1

j=1

and
sup Re l (Q(v1 ; : : : ; vm ))¡0
hold for l = 1; : : : ; d; whenever vj ∈ C and |vj |61 for j = 1; : : : ; m.
For the initial value problem of ODEs,

y(t)
˙ = f(t; y(t));

t¿0 and y(0) = y0 ;

a linear k-step method is given in a standard form as
k
X
j=0

j yn+j = h

k
X
j=0

j fn+j ;

(3)

G.-D. Hu, B. Cahlon / Journal of Computational and Applied Mathematics 102 (1999) 221–234

223

where h stands for the step-size and j , j are the formula parameters. Furthermore, a region RLM
ˆ
in the complex h-plane
is said to be the region of absolute stability if for all hˆ ∈ RLM the method
is absolutely stable [12].
Consider method (3) applied to system (2). Let tl = lh; l¿0; h¿0; and ul be the numerical
solution at the mesh points tl . We have
k
X

j un+j = h

j=0

k
X

j vn+j

(4)

[M u h (tn+j − )+N v h (tn+j − )]

(5)

j=0

and
vn+j = Lun+j +

m
X
=1

for n = 1; 2; : : : ; u h (t) = g(t) and t60, and u h (t) with t¿0 is dened by
u h (tl +h) =

s
X

Lˆj ()ul+j ;

j=−r

for 06¡1; l = 0; 1; : : : ; and
Lˆj () =

s
Y

(−k)
:
(j−k)
k=−r; k6=j

(6)

Hence,
u h (tn+j −l ) =

s
X

Lˆp (j )un+j−ll +p

(7)

Lˆp (j )vn+j−li +p ;

(8)

p=−r

and
vh (tn+j −i ) =

s
X

p=−r

where r; s¿0 are integers and r6s6r+2; lj = [j h−1 ]; j = lj −j h−1 ; 06j ¡1 for j = 1; : : : ; m,
lm ¿ · · · ¿l1 ¿s+1; here [q] denotes the smallest integer that is greater than or equal to q ∈ R.
A characterization of the region of absolute stability in NDDEs with multiple delays is given
by [6].
Lemma 2.2. If
(i) the assumptions of Lemma 2.1 hold, and
(ii) hl (Q(v1 ; : : : ; vm )) ∈ RLM for l = 1; : : : ; d and vj ∈ C such as |vj |61 for j = 1; : : : ; m;
(iii) r6s6r+2;
then the linear multistep method in (4)–(8) applied to (2) is asymptotically stable.

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G.-D. Hu, B. Cahlon / Journal of Computational and Applied Mathematics 102 (1999) 221–234

Next, we consider an application of sˆ-stage Runge–Kutta (RK in short) method in the ODE case
to system (2). Denote the stage values of the RK formula by kn;i . Let tl = lh, l¿0; h¿0, and ul
be the numerical solution at the mesh points tl . We obtain the natural RK scheme for system (2)
as follows:


kn;i = hL un +
un+1 = un +

sˆ
X

sˆ
X
j=1



aij kn; j +h

m
X
ˆ–=1



M ˆ– un−l ˆ– + ˆ– +

sˆ
X
j=1

aij kn−l ˆ– + ˆ–



+
;j

m
X

N ˆ– kn−l ˆ– + ˆ– ;i ;

(9)

ˆ–=1

bi kn;i :

(10)

i=1

Here i = 1; 2; : : : ; sˆ, aij and bi denote the parameters of the underlying Runge–Kutta method. For
n = 1; 2; : : : ; un−li +i = g((n−li + i )h) for n−li + i 60; un−li +i and kn−li +i ; j with n−li + i ¿0 are dened by the following respective interpolations:
un−li +i =

s
X

Lˆp (i )un−li +p ;

(11)

p=−r

kn−li +i ; j =

s
X

Lˆp (i )kn−li +p; j

(12)

p=−r

and
Lp () =

s
Y

(−k)
(p−k)
k=−r; k6=p

(13)

for 06¡1, i = 1; : : : ; m, j = 1; : : : ; sˆ, where r; s¿0 are integers, r6s6r+2 and li = [i h−1 ];  i =
li −i h−1 ; 06 i ¡1 for i = 1; : : : ; m; lm ¿ · · · ¿li ¿s+1. Here [q] denotes the smallest integer that is
greater than or equal to q ∈ R.
Let RRK denote the region of absolute stability of the RK method in the ODE case [12]. The
following are conditions for numerical stability of an explicit natural RK for system (2).
Lemma 2.3 (Hu and Hu [6]). Assume that
(i) the assumptions of Lemma 2.1 hold;
(ii) hi (Q(v1 ; : : : ; vm )) ∈ RRK for all i = 1; : : : ; d and vi ∈ C such as |vi |61 for i = 1; : : : ; m;
(iii) r6s6r+2.
Then the natural RK scheme in (9)–(13) for system (2) is asymptotically stable.
In view of Lemmas 2.2 and 2.3, the eigenvalues of Q(v1 ; : : : ; vm ) with |vi |61 govern the stability
of LM and RK methods. But it is dicult to select a stable step-size h by means of Lemmas 2.2 and
2.3 which require computation of i (Q(v1 ; : : : ; vm )), for i = 1; 2; : : : ; d and |vj |61 (j = 1; : : : ; m). In

G.-D. Hu, B. Cahlon / Journal of Computational and Applied Mathematics 102 (1999) 221–234

225

the following sections, on the basis of Lemmas 2.2 and 2.3, two simple estimations for the stability
regions for explicit LM and RK methods are derived by means of the logarithmic norm and the
spectral radius.
3. Estimation on numerically stable step-size for (2) via logarithmic norm
Lemma 3.1 (Lancaster and Tismenetsky [13]). Let W ∈ Cn×n . If (W )¡1; where (W ) is the spectral radius of the matrix W . Then (I +W )−1 exists and
(I +W )−1 = I −W +W 2 − · · · = I +(I +W )−1 (−W ):
Also; if kW k¡1 then
k(I + W )−1 k 6

1
:
1 − kW k

Let (W ) denote the logarithmic matrix norm, that is,
(W ) = lim+

→0

kI +
W k − 1
:



Lemma 3.2 (Desoer and Vidyasagar [2]). For each eigenvalue j (W ) of W ∈ Cd×d ; the inequality
−(−W )6Re j (W )6(W ) holds.
Denition 3.3. The real scalar quantities are dened as
X=

Pm

j=1

kNj Lk +
1−

E1 = −(−L) −

m
X

Pm

Pm
j=1 (
k=1
Pm
j=1 kNj k

kMj k − X;

j=1

E2 = min{0; l};
where l = (L) +

Pm

j=1

F1 = − (iL) −

kMj k + X ,

m
X

kMj k − X

j=1

and
F2 = (−iL) +

m
X
j=1

where i2 = −1.

kMj k + X;

kNj Mk k)

;

226

G.-D. Hu, B. Cahlon / Journal of Computational and Applied Mathematics 102 (1999) 221–234

Making use of these, we obtain the following estimations.
Theorem 3.4. Assume that the conditions of Lemma 2.1 hold. Then the eigenvalues of the matrix
Q(v1 ; : : : ; vm )(vj ∈ C and |vj |61) satisfy the following estimations:
E1 6Re j (Q(v1 ; : : : ; vm ))6E2
and
F1 6Im j (Q(v1 ; : : : ; vm ))6F2 :
Proof. From Section 2 we have


Q(v1 ; : : : ; vm ) = I −

m
X
j=1

−1 

Nj vj  L +

m
X
j=1



Mj v j 

for mj=1 kN k ¡ 1; vj ∈ C and |vj |61.
P
P
Let Nˆ = mj=1 Nj vj , and Mˆ = mj=1 Mj vj .
According to Lemma 3.2, we have the inequality
P

Re j (Q(v1 ; : : : ; vm ))6(Q(v1 ; : : : ; vm ))
for

Pm

j=1

|Nj | ¡1; vj ∈ C and |vj |61. In the following, we show that

(Q(v1 ; v2 ; : : : ; vm ))6(L) +

m
X

kMj k + X

j=1

holds. From Lemmas 3.1 and 3.2, for vj ∈ C and |vj |61, we obtain
(Q(v1 ; v2 ; : : : ; vm )) = [(I + Nˆ + Nˆ 2 + · · ·)(L + Mˆ )]
= [(L + Mˆ ) + (Nˆ + Nˆ 2 + · · ·)(L + Mˆ )]
6 (L) + (Mˆ ) + [(Nˆ + Nˆ 2 + · · ·)(L + Mˆ )]
6 (L) + kMˆ k + k(I + Nˆ + Nˆ 2 + · · ·)(Nˆ L + Nˆ Mˆ )k
6 (L) +

m
X

kMj k + (kI k + kNˆ k + kNˆ 2k + · · ·)(kNˆ Lk + kNˆ Mˆ k)

j=1

6 (L) +

m
X
j=1



kMj k + kI k +


m
X
j=1



2
m
X

kNj k +  kNj k + · · ·
j=1

G.-D. Hu, B. Cahlon / Journal of Computational and Applied Mathematics 102 (1999) 221–234



×

m
X

kNj Lk +

j=1

= (L) +

j=1

m
X

kMj k +

j=

= (L) +

m
X

m
X

m
X

(

k=1

Pm

j=1

227

!

kNj Mk k 

kNj Lk + mj=1 ( mk=1 kNj Mk k))
P
1 − mj=1 kNj k
P

P

kMj k + X = l:

j=1

Since the conditions of Lemma 2.1 hold, we have
Re j (Q(v1 ; v2 ; : : : ; vm )) 6 E2 = min{0; l}:
Thus Lemma 3.2 yields
−(−Q(v1 ; v2 ; : : : ; vm )) 6 Re j (Q(v1 ; v2 ; : : : ; vm ));
which in turn yields
E1 6 Re j (Q(v1 ; v2 ; : : : ; vm )):
Since
Im j (Q(v1 ; v2 ; : : : ; vm )) = Re j (−iQ(v1 ; v2 ; : : : ; vm ));
we obtain
−(iQ(v1 ; v2 ; : : : ; vm )) 6 Re j (−iQ(v1 ; v2 ; : : : ; vm )) 6 (−iQ(v1 ; v2 ; : : : ; vm )):
To demonstrate the inequality
F1 6 Im j (Q(v1 ; v2 ; : : : ; vm )) 6 F2 ;
we repeat similar calculations as for Re j (Q(v1 ; v2 ; : : : ; vm )). Thus the proof is completed.
Denition 3.5. Assume that
D(h) = hG;

Pm

j=1

kNj k ¡ 1. We dene

where h is the step-size and
G = {z = x + iy; E1 6x6E2 ; F1 6y6F2 }:
Now we state the main result which gives a simple way to nd a numerically stable step-size for
system (2).

228

G.-D. Hu, B. Cahlon / Journal of Computational and Applied Mathematics 102 (1999) 221–234

Theorem 3.6. Assume that the conditions of Lemma 2.1 and r6s6r + 2 hold.
(i) If D(h) ⊂ RLM for some positive h; then the explicit linear multistep method (4)–(8) for
system (2) is asymptotically stable.
(ii) If D(h) ⊂ RRK for some positive h; then the explicit natural Runge–Kutta method (9)–(13) is
asymptotically stable.
Proof. In the case of the LM method, due to Theorem 3.4, we have
hj (Q(v1 ; v2 ; : : : ; vm )) ⊂ D(h) ⊂ RLM ;
which implies the stability by virtue of Lemma 2.2. The proof for the RK is similar.

4. Estimation on numerically stable step-size for (2) via spectral radius
In this section we need the following denitions and lemmas. Let W ∈ Cn×n with elements wjk and
|W | denote the nonnegative matrix in Rn×n with elements |wj k|. Let W = {wjk } and V = {vjk } ∈ Rn×n .
We say |W |6V if and only if |wj k|6vjk for all pairs of ( j; k).
Lemma 4.1 (Lancaster and Tismenetsky [13]). Let W ∈Cn×n and V ∈Rn×n . If |W |6V; then (W )6
(V ); where (W ) and (V ) denote the spectral radii of W and V; respectively.
Denition 4.2. We dene


and

Y = I −
Yˆ = |L| +

m
X
j=1

m
X

−1 

|Nj |



m
X

|Nj L| +

j=1

|Mj | + Y:

j=1

Theorem 4.3. If

Pm

j=1

kNj k ¡ 1; then

(Q(v1 ; v2 ; : : : ; vm ))6(Yˆ )
for vj ∈ C and |vj |61;

j = 1; : : : ; m.

Proof. According to Lemma 4.1 and




m
X
j=1



|Nj |6

m
X
j=1

kNj k ¡ 1;

m
X
j=1

m
X
k=1

!

|Nj Mk | 

G.-D. Hu, B. Cahlon / Journal of Computational and Applied Mathematics 102 (1999) 221–234

229

we have


det I −

m
X
j=1



|Nj |6= 0:

Set Mˆ = Mj vj and Nˆ =
Nj vj where |vj |61, j = 1; : : : ; m. Thus |Mˆ |6
j=1 |Nj |. From Lemma 4.1, we obtain

Pm

P

P

Pm

j=1

|Mj | and |Nˆ | 6

|Q(v1 ; v2 ; : : : ; vm )| = |(I − Nˆ )−1 (L + Mˆ )|
= |(I + Nˆ + Nˆ 2 + · · ·)(L + Mˆ )|
6 |L| + |Mˆ | + (|Nˆ + Nˆ 2 + · · ·)(L + Mˆ )|
= |L| + |Mˆ | + |(I + Nˆ + Nˆ 2 + · · ·)(Nˆ L + Nˆ Mˆ )|
6 |L| +

m
X

|Mj | + (|I | + |Nˆ | + |Nˆ 2 | + · · ·)(|Nˆ L| + |Nˆ Mˆ |)

j=1

6 |L| +

m
X
j=1



m
X

|Mj | + |I |+

j=1



|Nj | + 

m
X
j=1




m
m
m X
X
X
×  |Nj L| + 
|Nj Mk |
j=1

= |L| +

m
X
j=1

= |L| +

m
X

2



|Nj | + · · ·

j=1 k=1



|Mj | + I −

m
X
j=1

−1 
m
m
X
X
|Nj |  |Nj L| +
j=1

|Mj | + Y = Yˆ :

j=1

m
X
k=1

!

|Nj Mk | 
(15)

j=1

According to Lemma 4.1, the proof is complete.
We need the following denition for the next result
Denition 4.4. We dene the region K(h) in the complex plane as
K(h) =

3

(
; ): 06
6h(Yˆ ); 66
;
2
2



′

′



where h is the step-size and (Yˆ ) is the spectral radius of the matrix Yˆ which is dened in
Denition 4.2.

230

G.-D. Hu, B. Cahlon / Journal of Computational and Applied Mathematics 102 (1999) 221–234

Theorem 4.5. Assume that the conditions of Lemma 2.1 and r6s6r + 2 hold.
(i) If K(h) ⊂ RLM for some positive h; then the explicit linear multistep method (4) – (8) for
system (2) is asymptotically stable for these choices of h.
(ii) If K(h) ⊂ RRK for some positive h; then the explicit natural Runge–Kutta method (9) – (13)
is asymptotically stable for these choices of h.
Proof. The proof is similar to the proof of Theorem 3.6.
Remark 4.6. Theorems 4.5 and Theorem 3.6 have shown a practical way to nd a stable step-size
h. Obviously, the choice of the step size h by hj (Q(v1 ; v2 ; : : : ; vm )) is sharper than the step size h
selected K(h) or D(h).

5. Examples
In this section we present several examples using the main results of this paper.
Consider system (2) with m = 2, i.e.,
u(t)
˙ = Lu(t) +

2
X

[Mj u(t − j ) + Nj u(t
˙ − j )];

t¿0

j=1

and
2 ¿ 1 ¿ 0:
Example 1. Let


−95



L=
 −1

0
−95

1



N1 = 


1
50
−1
25
1
25

2

−95

1
50
1
50
7
50




;

and


N2 = 


1
25
−1
50
3
50

1
10

0
1
25

3
50
3
50
2
25



0 
;

1

1
25
3
50
1
10






0

3

M1 = 
 2 −2



2

3

0

0

1

M2 = 
 1 −1




:

1

In this example kN1 k1 = 51 ; kN2 k1 = 15 .

4

6




3
;
1




2
;
5

(16)

G.-D. Hu, B. Cahlon / Journal of Computational and Applied Mathematics 102 (1999) 221–234

Fig. 1.

Fig. 2.

By direct calculation we obtain
E2 = −4;

E1 = −186;

F1 = 91;

F2 = −91:

Thus
G = {z = x + iy: −1866x6−4; −916y691}
and
D(h) = hG;

231

232

G.-D. Hu, B. Cahlon / Journal of Computational and Applied Mathematics 102 (1999) 221–234

Fig. 3.

and the stepsize h is determined by h ¡ 1 and hG ⊂ RLM or hG ⊂ RRK . We use the Adams–Moulton
method of order 4 (see [3] and Fig. 1). For system (2) we obtain the numerically stable step-size
to be h ¡ 0:01 and 0 ¡ h ¡ 1 .
In this example
(Yˆ ) = 136:0379
and
K(h) = {(
′ ; ): 06
′ 6136:0379h; 21 66 23 };
and a numerically stable step-size h is determined by h ¡ 1 and K(h) ⊂ RLM for a linear k-step
method applied to system (2) and K(h) ⊂ RRK for RK method applied to system (2). In the case
of the Adams–Bashforth method of order 3 applied to system (2), a numerically stable step-size is
determined by 0 ¡ h ¡ 1 and h ¡ 0:01; see [3].
Notice that if we use
(L) +

2
X
j=1

kMj k1

P2

j=1

kNj k1 kLk1 +
1−

P2

j=1

P2

j=1

P2

k=1

kNj k1

kNj k1 kMk k1

;

G.-D. Hu, B. Cahlon / Journal of Computational and Applied Mathematics 102 (1999) 221–234

233

as in [3], we get
(Q(v1 ; v2 ))61;
which is inconclusive, and asymptotic stability is not guaranteed.
Example 2. Consider system (2) again with the following matrices:
L=

"

−12

1

2

−12

1

0

M2 =

"

N2 =

"

1 −1

#

1
5

1
2

− 101

0

#

M1 =

N1 =

;

#

;

"

"

1
10
− 15

1

0

2

−2
1
5
3
10

#

#

;

;

:

In this example kN1 k1 + kN2 k1 = 1 and Theorems 3 and 5 cannot be applied. However, considering
the real parts of the eigenvalues of Q(v1 ; v2 ), we can see in Figs. 2 and 3 that
Re 1 (v1 ; v2 ) ¡ 0;

Re 2 (v1 ; v2 ) ¡ 0;

respectively. The surfaces are derived in this example from Maple. Therefore, system (16) with the
above matrices is asymptotically stable, and for h ¿ 0 suciently small, the linear multistep method
(4)–(8) and RK method (9)–(13) applied to (16) are asymptotically stable.
References
[1] C.T.H. Baker, Numerical analysis of Volterra functional and integral equations, in: I.S. Du, G.A.Watson (Eds.),
The State of the Art in Numerical Analysis, Oxford, 1997, pp. 193 – 222.
[2] C.A. Desoer, M. Vidyasagar, Feedback Systems: Input– Output Properties, Academic Press, New York, 1975.
[3] C.W. Gear, Numerical Initial Value Problems, Prentice-Hall, Eaglewood Clis, NJ, 1971.
[4] K.J. in’t Hout, Stability analysis of Runge – Kutta methods for systems of delay dierential equations, IMA J. Numer.
Anal. 17 (1997) 17 – 27.
[5] G.Da. Hu, Stability and numerical analysis for delay dierential systems, Ph.D. Thesis, Nagoya University, May
1996.
[6] G.Da. Hu, G.Di Hu, Stability properties of numerical methods for neutral delay dierential systems with multiple
delays, submitted.
[7] G.Da. Hu, T. Mitusi, Stability analysis of numerical methods for systems of neutral delay-dierential equations, BIT
35 (1995) 504 – 515.
[8] G.Da. Hu, T. Mitsui, G.Di. Hu, Simple estimations on numerically stable step-size for neutral delay-dierential
systems, Preprint series in Mathematical Sciences, School of Informatics and Sciences and Graduate School of
Human Informatics, Nagoya University, No. 22, April 1996.
[9] G.Di. Hu, G.Da. Hu, M.Z. Liu, Estimation of numerically stable step-size for neutral delay-dierential equations via
spectral radius, J. Compt. Appl. Math. 78 (1997) 311– 316.
[10] H.J. Tian, J.X. Kuang, The numerical stability of linear multistep methods for delay dierential equations with many
delays, SIAM J. Numer. Anal. 33 (3) (1996) 883 – 889.

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G.-D. Hu, B. Cahlon / Journal of Computational and Applied Mathematics 102 (1999) 221–234

[11] J.X. Kuang, J.X. Xiang, H.J. Tian, The asymptotic stability of one-parameter methods for neutral delay dierential
equations, BIT 34 (1994) 400 – 408.
[12] J.D. Lambert, Numerical Methods for Ordinary Dierential Systems, Wiley, New York, 1991.
[13] P. Lancaster, M. Tismenetsky, The Theory of Matrices, Academic Press, Orlando, FL, 1985.
[14] L.M. Li, Stability of linear neutral delay-dierential systems, Bull. Austral. Math. Soc. 38 (1988) 339 – 344.